Open Pseudohyperboloidal Resonator with an Annular Output Aperture: A Candidate New Geometry-Defined Class

A resonator is usually specified by the geometry of its reflecting elements and a separate choice of the opening through which radiation is released. We introduce an open pseudohyperboloidal (PHB) resonator: a pair of facing reflecting surfaces formed by revolving the interior branch of a radially shifted hyperbola, with an annular output aperture created by truncating one reflector. This construction is proposed as a potentially fundamentally new geometry-defined class within the wider family of open electromagnetic resonators, rather than as a new use of annular outcoupling alone. The parent conic induces external focal circles and, for a precisely specified meridional family, an exact ray-transport law that relates successive reflection points to the designed opening. The present work derives the three-dimensional boundary and aperture conditions, distinguishes this restricted geometrical theorem from the full wave problem, and reports source-off resonant modes obtained by solving Maxwell’s equations for several cavity shapes and azimuthal sectors. Matched nonfocal deformations test how the spectrum responds to loss of the exact generating conic. We situate the proposal relative to conventional stable, unstable, telescopic and annularly coupled resonators. The numerical evidence supports a reproducible passive PHB resonator family; the uniqueness of its full-wave mechanism and any general performance advantage remain separate research questions. The measured resonant frequencies and radial frequency responses show reproducible geometry-dependent differences relative to the volume-matched nonfocal control on both investigated grids; these observations do not establish an exclusive source-free modal mechanism. A curvature-based comparison also distinguishes the specified PHB boundary from ordinary circular-arc toroidal reflectors; neither observation is claimed to exclude every generalized toric or freeform predecessor. Within the broader GWE programme, separate ray-geometrical laws for pseudoparaboloidal and pseudoellipsoidal constructions motivate prospective additional geometrically organized open cavities; no electromagnetic resonator validation of those two candidates is asserted here.